Journal of Operator Theory

Volume 64, Issue 2, Fall 2010 pp. 299-319.

$C^\ast$-algebras of tilings with infinite rotational symmetry

Authors: Michael F. Whittaker
Author institution: Mathematics and Statistics, University of Victoria, PO BOX 3060, STN CSC Victoria B.C., Canada V8W 3R4; Current address: School of Mathematics and Applied Statistics, University of Wollongong, WOLLONGONG NSW 2522, Australia

Summary: A tiling with infinite rotational symmetry, such as the Conway--Radin Pinwheel Tiling, gives rise to a topological dynamical system to which an \etale equivalence relation is associated. A groupoid $C^\ast$-algebra for a tiling is produced and a separating dense set is exhibited in the $C^\ast$-algebra which encodes the structure of the topological dynamical system. In the case of a substitution tiling, natural subsets of this separating dense set are used to define an $\AT$-subalgebra of the $C^\ast$-algebra. Finally our results are applied to the Pinwheel Tiling.

Keywords: $C^\ast$-algebras, dynamical systems, noncommutative geometry, operator algebras, substitution tilings, tilings

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